Related papers: Characteristic classes of proalgebraic varieties a…

Let G be a linear algebraic group over a field F and X be a projective homogeneous G-variety such that G splits over the function field of X. In the present paper we introduce an invariant of G called J-invariant which characterizes the…

Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties…

The power structure over the Grothendieck (semi)ring of complex quasi-projective varieties constructed by the authors is used to express the generating series of classes of Hilbert schemes of zero-dimensional subschemes on a smooth…

We introduce the notion of a majority category --- the categorical counterpart of varieties of universal algebras admitting a majority term. This notion can be thought to capture properties of the category of lattices, in a way that…

This article is the continuation of [LS12]. We use categories of matrix factorizations to define a morphism of rings (= a Landau-Ginzburg motivic measure) from the (motivic) Grothendieck ring of varieties over $\mathbb{A}^1$ to the…

The proalgebraic fundamental group of a connected topological space $X$, recently introduced by the first author, is an affine group scheme whose representations classify local systems of finite-dimensional vector spaces on $X$. In this…

We prove a formula, originally due to Feit and Fine, for the class of the commuting variety in the Grothendieck group of varieties. Our method, which uses a power structure on the Grothendieck group of stacks, allows us to prove several…

We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period…

We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the "Euler characteristic integral" of a certain cohomotopy class over its scheme of…

We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for a certain type of Courant algebroids, called…

Motivic equivalence for algebraic groups was recently introduced in [9], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms…

We establish a formula for the classes of certain tori in the Grothendieck ring of varieties, in terms of its lambda-structure. More explicitly, we will see that if L* is the torus of invertible elements in the n-dimensional separable…

The geometric motivic Poincar\'e series of a variety, which was introduced by Denef and Loeser, takes into account the classes in the Grothendieck ring of the sequence of jets of arcs in the variety. Denef and Loeser proved that this series…

Let k be a field of characteristic not 2. We conjecture that if X is a quasi-projective k-variety with trivial motivic Euler characteristic, then Sym$^n$X has trivial motivic Euler characteristic for all n. Conditional on this conjecture,…

The relative Grothendieck group $K_0(\m V/X)$ is the free abelian group generated by the isomorphism classes of complex algebraic varieties over $X$ modulo the "scissor relation". The motivic Hirzebruch class ${T_y}_*: K_0(\m V /X) \to…

We construct invariants of birational maps with values in the Kontsevich--Tschinkel group and in the truncated Grothendieck groups of varieties. These invariants are morphisms of groupoids and are well-suited to investigating the structure…

We show that the zero-dimensional part of the pro-Chern-Schwarz-MacPherson class defined by Aluffi is equal to the pro-characteristic class in limit Borel-Moore motivic homology. A similar construction also produces a quadratic refinement…

We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski-Kazhdan style motivic integration for certain type of non-archimedean o-minimal fields, namely power-bounded T-convex valued…

We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series…

A remarkable result of Peter O'Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville's…